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The chi-square test appears in a lot of different places. Some recent data on Astrology, published in the May newsletter of the Skeptic Society, offers an interesting opportunity to show one of these tests. In an article offering dark matter as a possible explanation of the effects of astrology, the London Times published a list of the 1,000 richest people and their star signs. It noted a significant difference between the number born under Gemini and the number born under Pisces. A careful look at the full data set, though, shows that numbers observed could easily be explained by sampling error.
There are actually 1,067 people listed here, which suggests either a typographical error or perhaps inflation has hit the top 1,000 list. The question is whether the distribution of star signs is uniform or not. A non-uniform distribution could be taken as evidence for Astrology, but it would probably need replication to be taken seriously.
If the distribution is uniform, we would expect to see about 89 people in each star sign. We compare the expected count to the observed count using the Chi-square goodness of fit statistic, X2:
where Oi are the observed counts (110, 104, 95, ..., 73) and Ei are the expected counts (88.92 for each star sign).
The table below shows the calculation of
You should compare this statistic to a Chi-squared distribution with 11 degrees of freedom. The 95th percentile of this distribution is 19.7. Since your test statistic is smaller than 19.7, you would accept the null hypothesis and conclude that the distribution is uniform. Actually, you should state the conclusion more cautiously: there is insufficient evidence for a non-uniform distribution of star sign among the richest 1,000 people. You can also compute a p-value for this test statistic. In Microsoft Excel, the function CHIDIST(13.57,11) produces a p-value of 0.258.
Now if you just took the largest group, Gemini, and compared it to the smallest group, Pisces, you would indeed get a statistically significant differrence:
Compare this test statistic to a Chi-squared distribution with 1 degree of freedom, where the 95th percentile is 3.84. The p-value (CHIDIST(7.48,1)) is 0.0062.but there was no reason to believe a priori that Gemini would have a surplus of rich people compared to Pisces. Here's what the astrology.com web site says about Gemini:
Gemini is the third Sign of the Zodiac, and those born under this Sign will be quick to tell you all about it. That's because they love to talk! It's not just idle chatter with these folks, either. The driving force behind a Gemini's conversation is their mind. The Gemini-born are intellectually inclined, forever probing people and places in search of information. The more information a Gemini collects, the better. Sharing that information later on with those they love is also a lot of fun, for Geminis are supremely interested in developing their relationships. Dalliances with these folks are always enjoyable, since Geminis are bright, quick-witted and the proverbial life of the party. Even though their intellectual minds can rationalize forever and a day, Geminis also have a surplus of imagination waiting to be tapped. Can a Gemini be boring? Never!
and about Pisces
Pisces is the twelfth Sign of the Zodiac, and it is also the final Sign in the Zodiacal cycle. Hence, this Sign brings together many of the characteristics of the eleven Signs that have come before it. Pisceans, however, are happiest keeping many of these qualities under wraps. These folks are selfless, spiritual and very focused on their inner journey. They also place great weight on what they are feeling. Yes, feelings define Pisceans, and it's not uncommon for them to feel their own burdens (and joys) as well as those of others. The intuition of the Pisces-born is highly-evolved. Many people associate Pisceans with dreams and secrets, and it's a fair association, since those born under this Sign feel comfortable in an illusory world.
If you tried very hard, you could possibly infer some hints that Geminis are richer because they are smarter, and Pisces are poorer because of their selfless nature. I don't see anything in the description that would make me think so, and it is easy to develop post hoc rationalizations. Unless you had a hypothesis developed prior to data collection, you need to make some sort of adjustment.
The simplest adjustment is a Bonferroni correction. This adjusts the p-value by multiplying it by the number of possible comparisons that could be made. With 12 Astrology signs, there are 12*11/2=66 possible comparisons. The adjusted p-value would be 0.41 which is not statistically significant. Other adjustments are actually better than Bonferroni here, but they would lead to the same conclusion.
This is hardly a definitive study, but it does tend to support most of the other research on Astrology. The most prominent claim in support of Astrology is the Mars Effect, which was a claim by Michel Gauquelin in 1955 that prominent sports figures were more likely to be born at times when the planet Mars was in a certain position in the sky. A careful study of this is actually quite difficult because birth dates and times are not quite uniformly distributed and because it is hard to define what a prominent sport figure is. Jan Willem Nienhuys published a critical review of the Mars effect that appeared in the November/December 1997 issue of Skeptical Inquirer.
A good resource about Astrology is in the Skeptic's Dictionary website: www.skepdic.com/astrolgy.html.